238 
:\IR. S. S. HOUGH ON THE APPLICATION OF HAR^IONIC 
§ 11. Forced Tides. 
Leaving now the problem of the free oscillations, let us return to the equations of 
§ 5, when wm retain the y’s. It is obvious in the first place that a disturbing force 
whose potential at the surface is expressible by surface-harmonics of even order alone, 
or of odd order alone, will give rise to a forced oscillation of like character. Further 
we may consider separately the effects of the different terms in the disturbing 
potential and superpose the results. Suppose for example that the surface-value of 
the disturbing potential is expressible by the single harmonic term 
y,P„ (^) 
wLere w’e will suppose n even. 
The equations (23) w'hich determine the type may be written 
- + CJ7 .9 = 0 
C./5 . 7 - + C,./l 1.13 = 0 
L\_J{2n - 3) {2n - i) - CX. + 0,J{2n + 3) {2n + 5) = yX/IojuF 
C.„l{2n + l) {2n +3) — + ^,,+4./ (2n 7) (2/i -b 9) = 0 
w’ith the condition that = 0 ; whence w^e obtain 
Cr.jd = (2r - 3) (2r - 1) {r < n + 1) 
CrJCr = (2r + 3) (2r -f 5) K ,+2 - 1) 
and therefore 
- L„} = y„/l/4a;%U 
or 
_ 
yF 
^.orcc^ (H„_ 
-2 + F «+2 L,-,) 
Thus the height of the tide is given by 
. . . -f (2^1 - 1) (2r. - 3) (2n - 5) {2n - 7) 
+ (2p — l){2;z — 3)H„_2P«_o-bP;,+(2?^ ■l"3)(2/i-b5)K„4.2pw+2 
-p (2n + 3 ) {2n + 5) [2n -p 7)(2n + 9) + ... 
4iO}~a- (Hi,_2 -p — L„) 
The expressions H, K, L all depend on X the frequency of the disturbing force. It 
is obvious from the above that the tides become very large when X approaches a root 
of the equation 
L,( “* K„4.o ~ 0 ^ 
